Question

points q, r and s all lie on line p where r is the mid - point and q and p are the endpoints. if qr = 2x² - 36 and rp = 3x, determine the length of qp. (a) 18 units (b) 24 units (c) 9 units (d) 36 units.
*#2.) ab is the perpendicular bisector of qp. the endpoints of qp are q(3,2) and p(-1,0). what is the y - intercept of ab? the use of the grid is optional. (a) 0 (b) 1 (c) - 1 (d) - 2
#3.) aaron and beth live on highway 61 and the intersection for interstate 87 is halfway between their homes. if the coordinates of aarons home are (5, - 5) and beths are (-3,1) what point represents the intersection of highway 61 and interstate 87? the use of the grid is optional. (a) (0, - 6) (b) (4, - 3) (c) (2, - 4) (d) (1, - 2)

Explanation:

Step1: Find mid - point formula

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.

Step2: Calculate x - coordinate of mid - point for third problem

For points $(x_1 = 5,y_1=-5)$ and $(x_2=-3,y_2 = 1)$, the x - coordinate of the mid - point is $\frac{5+( - 3)}{2}=\frac{2}{2}=1$.

Step3: Calculate y - coordinate of mid - point for third problem

The y - coordinate of the mid - point is $\frac{-5 + 1}{2}=\frac{-4}{2}=-2$. So the mid - point (intersection point) is $(1,-2)$.

For the second problem:

Step1: Find mid - point of QP

The mid - point of $Q(3,2)$ and $P(-1,0)$ is $(\frac{3+( - 1)}{2},\frac{2 + 0}{2})=(1,1)$.

Step2: Find slope of QP

The slope of the line passing through $Q(3,2)$ and $P(-1,0)$ is $m_{QP}=\frac{2-0}{3-( - 1)}=\frac{2}{4}=\frac{1}{2}$.

Step3: Find slope of perpendicular bisector AB

The slope of the perpendicular bisector $AB$ of $QP$ is $m_{AB}=-2$ (since the product of slopes of two perpendicular lines is - 1).

Step4: Use point - slope form to find equation of AB

Using the point - slope form $y - y_1=m(x - x_1)$ with $(x_1,y_1)=(1,1)$ and $m=-2$, we have $y - 1=-2(x - 1)$.

Step5: Find y - intercept

Expand the equation: $y-1=-2x + 2$, then $y=-2x+3$. The y - intercept is 3.

Answer:

Third problem:

D. $(1,-2)$

Second problem:

B. 3